Infinite Interval Problems for Differential, Difference and Integral Equations

349 Pages·2001·12.31 MB·English

by  Ravi P. Agarwal

INFINITE INTERVAL PROBLEMS FOR DIFFERENTIAL,
DIFFERENCE AND INTEGRAL EQUATIONS
Infinite Interval Problems
For Differential,
Difference and Integral Equations
by
Ravi P. Agarwal
National University 0/ Singapore.
Singapore. Republic 0/ Singapore
and
DonalO'Regan
University o/Ireland.
Galway, Ireland
Springer-Science+Business Media, B.V
Lihrary ofD:mgress Cataloging-in-Publication Data
Agaiwal, Ravi P.
Infinite interval problems for differe:ntial, diffcrence, aud integral equations I Ravi P.
Agarwal arul Dona! O'Regan.
p.cm.
Includes bibliographicai references arul index.
ISBN 978-94-010-3834-8 ISBN 978-94-010-0718-4 (eBook)
DOI 10.1007/978-94-010-0718-4
1. Boundary value problems.--Numerical solutions. 2. Difference equations--Num.erical
n.
solutions. 3. Integral equations--Numerica1 solutions. I. O'Regan, Dona!. Title.
QA379 .A348 2001
SIS'.3S--dc21
ISBN 978-94-010-3834-8 2001029534
Printed on acid{ree paper
AII Rights Rescrved
© 2001 Springer Science-Business Media I10rdrecht
Originally publistlcd by Kluwcr Academic Publishcrs in 2001
Softcover reprint ofthe hardcover 1 st edition 2001
No part of the material protected by this copyright notice may be reproduced or
utili..:ed in any fonn ar by any means, elecuonic or mechanical,
including photocopying, reoording ar by any informatian storage and
retrieval system, WithOllt written permission from thc copyright owner.
Contents
Preface ix
Chapter 1 Second Order Boundary Value Problems 1
1.1. Introduction 1
1.2. Some Examples 3
1.3. Preliminary Results 7
1.4. Existence Theory for Problems (1.1.1) - (1.1.3) 10
1.5. Existence Theory for Problems of Type (1.1.4) 16
1.6. Existence Theory for Problems of Type (1.1.5) 23
1.7. Existence Theory for Problems of Type (1.1.6) 30
1.8. Existence Theory for Problems of Type (1.1.7) 40
1.9. Existence Theory for Problems of Type (1.1.8) 48
1.10. Existence Theory for Problems of Type (1.1.9) 54
1.11. Existence Theory for Singular Problems 58
of Type (1.1.1) - (1.1.3)
1.12. Existence Theory for Problems (1.1.10) and (1.1.11) 79
1.13. Notes and Remarks 84
1.14. References 85
Chapter 2 Higher Order Boundary Value Problems 90
2.1. Introduction 90
2.2. Preliminary Results 90
2.3. Existence Theory for Conjugate Type Problems 92
2.4. Existence Theory for Right Focal Type Problems 103
2.5. Notes and Remarks 107
2.6. References 108
vi
Chapter 3 Continuous Systems 110
3.1. Introduction 110
3.2. Linear Problems (3.1.3), (3.1.2) 111
3.3. Nonlinear Problems (3.1.1), (3.1.2) 117
3.4. Nonlinear Problems (3.1.4), (3.1.2) 130
3.5. Nonlinear Problems (3.1.5), (3.1.6) 132
3.6. Notes and Remarks 136
3.7. References 136
Chapter 4 Integral Equations 139
4.1. Introduction 139
4.2. Existence Theory for (4.1.1) and (4.1.2) 141
4.3. Existence Theory for (4.1.3) and (4.1.4) 149
4.4. Existence Theory for (4.1.5) 157
4.5. Existence Theory and Behaviour 160
of Solutions to (4.1.6)
4.6. Existence Theory for (4.1.7) and (4.1.8) 179
4.7. Existence and Approximation for (4.1.9) 187
4.8. Abstract Volterra Equations 200
4.9. Periodic and Almost Periodic Solutions to (4.1.10) 210
4.10. Periodic Solutions to (4.1.11) 220
4.11. Notes and Remarks 228
4.12. References 229
Chapter 5 Discrete Systems 233
5.1. Introduction 233
5.2. Linear Problems (5.1.3), (5.1.2) 234
5.3. Nonlinear Problems (5.1.1), (5.1.2) 239
5.4. Nonlinear Problems (5.1.4), (5.1.2) 253
5.5. Second Order Problems (5.1.5), (5.1.7) 256
5.6. Summary Discrete Systems (5.1.8) 258
5.7. Urysohn Discrete Equations (5.1.9) 266
5.8. Notes and Remarks 274
5.9. References 275
vii
Chapter 6 Equations in Banach Spaces 277
6.1. Introduction 277
6.2. Continuous Equations 278
6.3. Discrete Equations 283
6.4. Continuous and Discrete Equations 288
6.5. Notes and Remarks 292
6.6. References 292
Chapter 7 Multivalued Equations 294
7.1. Introduction 294
7.2. Existence Theory for (7.1.1) 295
7.3. Solution Set of (7.1.2) 309
7.4. Existence Theory for (7.1.3) 314
7.5. Existence Theory for (7.1.4) and (7.1.5) 318
7.6. Notes and Remarks 325
7.7. References 326
Chapter 8 Equations on Time Scales 329
8.1. Introduction 329
8.2. Existence Theory for (8.1.1) 330
8.3. Notes and Remarks 337
8.4. References 337
Subject Index 339
Preface
Infinite interval problems abound in nature and yet until now there
has been no book dealing with such problems. The main reason for this
seems to be that until the 1970's for the infinite interval problem all the
theoretical results available required rather technical hypotheses and were
applicable only to narrowly defined classes of problems. Thus scientists
mainly offer~d and used special devices to construct the numerical solution
assuming tacitly the existence of a solution. In recent years a mixture of
classical analysis and modern fixed point theory has been employed to study
the existence of solutions to infinite interval problems. This has resulted
in widely applicable results. This monograph is a cumulation mainly of
the authors' research over a period of more than ten years and offers easily
verifiable existence criteria for differential, difference and integral equations
over the infinite interval. An important feature of this monograph is that
we illustrate almost all the results with examples.
The plan of this monograph is as follows. In Chapter 1 we present
the existence theory for second order boundary value problems on infinite
intervals. We begin with several examples which model real world phenom
ena. A brief history of the infinite interval problem is also included. We
then present general existence results for several different types of boundary
value problems. Here we note that for the infinite interval problem only
two major approaches are available in the literature. The first approach is
based on a clever diagonalization process whereas the second is based on
the Furi-Pera fixed point theorem. Chapter 2 establishes existence theory
for higher order differential equations together with conjugate (Hermite)
and right focal (Abel-Gontscharoff) type boundary data over the infinite
interval. In Chapter 3 we provide an existence theory for continuous sys
tems over the infinite interval. For the linear problem we give necessary and
sufficient conditions for the existence of a solution. For the nonlinear prob
lem besides sufficient conditions some iterative methods are also discussed.
Chapter 4 presents a systematic existence theory for integral equations of
Volterra and Fredholm type over the infinite interval. Here the existence
x
of solutions in various spaces is addressed. We also introduce the notion of
collectively compact operators and strict convergence to establish the exis
tence and approximation of solutions to some nonlinear operator equations
on the infinite interval. In addition the solution set of abstract Volterra,
functional and functional differential equations in different spaces is dis
cussed, and applications to integral and integrodifferential equations and
initial value problems are given. Finally, in this chapter we establish a
variety of existence results which guarantee the existence of periodic and
almost periodic solutions to some nonlinear integral equations over the en
tire real line. Some of these results apply directly in modeling the spread
of infectious diseases. In Chapter 5 we study discrete systems over the infi
nite interval. Here discrete analogues of several results established in earlier
chapters are presented. An existence theory for summary discrete systems
and nonlinear Urysohn type discrete equations is also offered. Chapter 6
presents general existence principles for nonlinear integral equations and
their discrete analogues in real Banach spaces over the infinite interval. In
Chapter 7 we first discuss the existence of one (or more) solutions to non
linear integral inclusions. Then we investigate the topological structure of
the solution set of Volterra integral inclusions. Next existence criteria for
Fredholm integral inclusions is presented. We conclude this chapter with
an existence theory for abstract operator inclusions, where the operators
involved are of upper semicontinuous or lower semicontinuous type. Chap
ter 8, which is our final chapter, presents an existence theory for second
order time scale boundary value problems over the infinite interval. Here we
establish two types of results, the first one is based on a growth argument,
whereas the second is based on an upper and lower solution type argument.
We hope this monograph is timely and will fill the vacuum in the litera
ture on the existence theory of differential, difference and integral equations
over the infinite interval. We also hope that it will stimulate further re
search and development in this important area.
It is impossible to acknowledge individually colleagues and friends to
whom we are indebted for assistance, inspiration and criticism during the
preparation of this monograph. We must, however, express our appreciation
and thanks to Maria Meehan for her collaboration in research, and Sadhna
for her careful typing of the entire manuscript.
Ravi P Agarwal
Donal 0 'Regan
Chapter 1
Second Order
Boundary Value Problems
1.1. Introduction
This chapter presents existence theory for second order boundary value
problems on infinite intervals. There are two major approaches in the lit
erature to establish existence of solutions to boundary value problems on
infinite intervals. The first approach is based on a diagonalization process
whereas the second is based on the Furi-Pera fixed point theorem. Both
approaches will be presented in this chapter. In Section 1.2 we list sev
eral examples from the real world phenomena which motivate the study of
boundary value problems on infinite intervals. In Section 1.3 we discuss
some infinite interval problems which date back to 1896 and state several
fundamental results which are well known. In Section 1.4 we examine the
problems
x" + ¢(t)f(t, x, Xl) = 0, 0 < t < 00
{ (1.1.1)
x(O) = 0, x(t) bounded on [0,00),
x" + ¢(t)f(t, x, Xl) = 0, 0 < t < 00
{ (1.1.2)
x(O) = 0, limHoo x(t) exists,
and
XII + ¢(t)f(t, x.' Xl) =~' 0 < t < 00
{ (1.1.3)
x(O) = 0, IImHoo x (t) = O.
By putting physically reasonable assumptions on ¢ and f we will show
that (1.1.1) - (1.1.3) have solu°tio ns x E Gl[O, 00) nG2(0, 00) with x(t) >
0, t E (0,00) even if x(t) == is also a solution. In Sections 1.5 and 1.6,
respectively, we establish the existence of nonnegative solutions to problems
R. P. Agarwal et al., Infinite Interval Problems for Differential, Difference and Integral Equations
© Springer Science+Business Media Dordrecht 2001
2 Chapter 1
of the type
~tt) (p(t)x')' ,= </J(t)f(t,x,p(t)x'), O<t<oo
{ (1.1.4)
hmt---+o+ p(t)x (t) = 0
x(t) bounded on [0,00),
and
ptt) (p(t)x')' = </J(t)f(t,x,p(t)x'), O<t<oo
{ (1.1.5)
-o:x(O) + ,8limt---+o+ p(t)x'(t) = c, 0: > 0, ,8 2:: 0, c::; 0
x(t) bounded on [0,00).
We discuss in Section 1.7 a special case of (1.1.5), namely
ptt) (p(t)x')' = </J(t)f(t,x,p(t)x'), 0 < t < 00
{ (1.1.6)
x(O) = 0
x(t) bounded on [0,00).
An existence theory motivated by the upper and lower solution approach of
Jackson (see Theorems 1.3.8 and 1.3.9) will be presented here. In Sections
1.8 and 1.9, respectively, we employ more sophisticated analysis to present
the existence of solutions to problems of the type
_I (p(t)x')' = </J(t)f(t, x,p(t)x'), O<t<oo
p(t)
{ (1.1.7)
limt---+o+ p(t)x'(t) = 0
limt---+oo x(t) = 0,
and
ptt) (p(t)x')' = </J(t)f(t, x,p(t)x'), O<t<oo
{ - o:x(O) + ,8limt---+o+ p(t)x'(t) = c, 0: > 0, ,8 2:: 0, c::; 0 (1.1.8)
limt---+oo x(t) = O.
We discuss in Section 1.10 a special case of (1.1.8), namely
_1_(p(t)x')' = </J(t)f(t,x,p(t)x'), 0 < t < 00
p(t)
{
(1.1.9)
x(O) = 0
limt---+oo x(t) exists.